A technique relating to the present invention is disclosed, for example, in the following Patent Document 1. In the Patent Document 1, there is disclosed a compression self-ignition engine configured to supply auxiliary energy to an air-fuel mixture in a combustion chamber to promote self-ignition, wherein a target in-cylinder temperature at top dead center of a compression stroke is set, and an in-cylinder temperature is controlled to attain the target in-cylinder temperature, by means of an intake air heating device provided in an intake passage